Word Count : 299

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Assume a power series about $x=0$: $$ y=\sum_{n=0}^\infty a_n x^n,\qquad y'=\sum_{n=1}^\infty n a_n x^{n-1},\qquad y''=\sum_{n=2}^\infty n(n-1)a_n x^{n-2}. $$ Substitute into $(x+2)y''+x y' - y=0$. Split $(x+2)y''=x y''+2y''$ and expand each term with indices aligned to powers $x^m$. $x y''=\sum_{n=2}^\infty n(n-1)a_n x^{n-1}=\sum_{m=1}^\infty (m+1)m a_{m+1} x^{m}.$ $2y''=\sum_{n=2}^\infty 2n(n-1)a_n x^{n-2}=\sum_{m=0}^\infty 2(m+2)(m+1)a_{m+2} x^{m}.$ _________ _______ ______ _____ ____ ______ _____ __________ _____ _______.
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